#7566816

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24354	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1809 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(19,26)(22,27)(23,32)(28,31);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,27)(20,24)(21,30)(22,26)(23,28)(25,29)(31,32);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,22)(20,26)(21,28)(23,25)(24,27)(29,31)(30,32);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,19)(18,23)(20,26)(21,31)(22,25)(24,32)(27,30)(28,29);(2,6)(7,8)(10,14)(13,15)(19,26)(22,27)(23,32)(28,31)

Polycyclic presentation:

2^9|a^2=gi,c^2=fgi,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257656, #72264023, #72264250
maximal normal subgroup of	#72257656, #72264023, #72264250
maximal quotient of	#72308317

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7474012);